The most important theorem in differential geometry: Gauss-Bonnet theorem

To try everything Brilliant has to offer—free—for a full 30 days, visit https://brilliant.org/Mathemaniac/. You’ll also get 20% off an annual premium subscription. This video was sponsored by Brilliant. Boundary term: https://youtu.be/Tf7VwAIQCSg Previous second channel video on spherical triangle area: https://www.youtube.com/watch?v=bAEVp2uBAxY You might also realise that a lot of the arguments are similar to the one that I mentioned here: https://www.youtube.com/watch?v=7Ju9f9odKX4 This is a proof (technically two?) taken from Tristan Needham's Visual Differential Geometry and Forms. Files for download: Go to https://www.mathemaniac.co.uk/download and enter the following password: gaussbonnet "The Gauss-Bonnet theorem... is purely a theorem of differential geometry, arguably the most fundamental and important one of all." This is taken from p.8 of John M. Lee's Riemannian Manifolds, a quote pointed out to me by my friend Eugene So. If you ask for my favourite theorem a year ago, I would have answered Theorema Egregium first and Gauss-Bonnet a close second. You can even testify this by going to my Q & A video: https://youtu.be/hRTBVrEkb6A?feature=shared&t=762 However, because of my exposure to more physics, I started to love Gauss-Bonnet a lot more. The major reason is that Theorema Egregium is only applicable to 2D surfaces, where extrinsic curvature = intrinsic curvature, in fact that's the statement of the theorem. However, the Gauss-Bonnet theorem can be recast into an intrinsic language, using Theorema Egregium, which is more useful and generalisable to higher dimensions. This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject. Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels: https://forms.gle/QJ29hocF9uQAyZyH6 If you want to know more interesting Mathematics, stay tuned for the next video! SUBSCRIBE and see you in the next video! If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos. Video chapters: 00:00 Introduction 01:44 Gaussian curvature 04:36 Intuition (too hand-wavy) 07:23 Main idea 08:06 Parallel transport, geodesics, holonomy 13:35 Gauss map preserves parallel transport 15:40 Adding up local contributions 19:15 Generalisations Social media: Facebook: https://www.facebook.com/mathemaniacyt Instagram: https://www.instagram.com/_mathemaniac_/ Twitter: https://twitter.com/mathemaniacyt Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!) Merch: https://mathemaniac.myspreadshop.co.uk Ko-fi: https://ko-fi.com/mathemaniac [for one-time support] For my contact email, check my About page on a PC. See you next time!